# Let an object X of mass m be placed on a slope inlined at an angle theta to the horizontal. Let u=coefficient of static friction . 1. if theta > angle of repose then find the maximum and minimum...

Let an object X of mass m be placed on a slope inlined at an angle theta to the horizontal.

Let u=coefficient of static friction .

1. if theta > angle of repose then find the maximum and minimum force that can be applied to prevent the object from slipping.

2. if theta < angle of repose then find the minimum and maximum force that can be applied to keep the body stationary.

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The figure with all possible cases is below.

Let ` ` `theta` be the angle of inclined and `beta ` be the angle of repose.

The angle of repose of a inclined plane is the maximum angle for which the body does not slip.

`G_p = F_f` , or `m*g*sin(beta) =miu*m*g*cos(beta)`

which means

`beta = arctan(miu)`

1. `theta > arctan(miu)`

Case a) the mass is slipping down the plane. To prevent slipping

`F > m*g*sin(theta) -miu*m*g*cos(theta) = F_(min)`

Case b) the mass is slipping up the plane. To prevent slipping

`F < m*g*sin(theta) +miu*m*g*cos(theta) =F_(max)`

2. `theta < arctan(miu)`

Case c) the friction force is directed upwards. To keep the mass stationary

`F < miu*m*g*cos(theta) -m*g*sin(theta) = F_(max)`

Case d) the friction force is directed downwards. To keep the mass stationary

`F > -(m*g*cos(alpha) +m*g*sin(alpha)) = F_(min)`